ESSENTIAL UNDERSTANDING ANDOBJECTIVES Essential Understanding: The graph of any quadratic function is the transformation of the graph of the parent function y = x2 Objectives: Students will be able to: Identify and graph quadratic functions Identify and graph the transformations of quadratic functions (reflect, stretch, compression, translation) Solve for the minimum and maximum values of parabolas
IOWA CORE CURRICULUM Algebra A.CED.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Functions F.IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F.IF.7. Graph functions expressed symbolically, and show features of the graph, by hand in simple cases and using technology for more complicated cases. F.BF.3. Identify the effect on the graph of f(x) + k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
VOCABULARY Parabola: the graph of a quadratic function, it makes a U shape Quadratic Function: ax2 + bx + c Vertex Form: f(x) = a(x – h)2 +k, where a doesn’t equal zero, vertex is (h, k) Axis of Symmetry: line that divides the parabola into two mirror images. Equation x = h Parent Function: y = x2
GRAPHING A QUADRATIC FUNCTION Graphing a Function in the form f(x) = ax2 f(x) = (1/2)x2 Plot the vertex Find and plot two points on one side of the axis of symmetry Plot the corresponding points on other side of the axis of symmetry Sketch the curve Graph: f(x) = -(1/3)x2 What can you say about the graph of the function f(x) = ax2 if a is a negative number?
TRANSFORMATIONS Vertex form: f(x) = a(x-h)2 + k Reflection: if a is positive the graph opens up, if a is negative it reflects across the x-axis and opens downward If the parabola opens upward, the y coordinate of the vertex is a minimum If the parabola opens downward, the y coordinate of the vertex is a maximum Stretch a > 1 the graph becomes more narrow Compression 0< a < 1 the graph becomes more flat
TRANSFORMATIONS Standard form: f(x) = a(x-h)2 + k Vertical Translation: k value, on the outside of the parentheses. Moves graph up and down Horizontal translation: opposite of the h value, on the inside of the parentheses. Moves graph left and right.
EXAMPLES For the equations below, write the vertex, the axis of symmetry, the max or min value, and the domain and range. Then describe the transformations. f(x) = x2 – 5 f(x) = (x – 4)2 f(x) = -(x + 1)2 f(x) = 3(x – 4)2 – 2 f(x) = -2(x +1)2